Variable filter

ABSTRACT

A variable filter has a signal loop defined between a signal input and a signal output. A plurality of circuit elements connected in the signal loop, the plurality of circuit elements comprising a frequency tunable resonator, and an adjustable scaling block that applies a gain factor that is adjustable in a range that comprises a positive gain and a negative gain. A controller is connected to 1) tune the frequency tunable resonator; and to 2) adjust the gain factor of the adjustable scaling block between a negative gain factor to a positive gain factor providing for variable Q independent of frequency.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/360,813, filed Nov. 23, 2016, which claims benefit of U.S.Provisional Patent Application No. 62/258,867, filed Nov. 23, 2015, allof which is incorporated by reference herein in their entirety.

TECHNICAL FIELD

This relates to a variable filter that may be used for analog rf (radiofrequency), microwave and millimeter wave frequency filteringapplications, and may also be extended to higher or lowerelectromagnetic frequencies. The variable filter may be controllable andapplicable in band-pass frequency filtering applications in which it isdesirable to be able to electronically adjust the pass-band centerfrequency, the bandwidth, or both. The variable filter may be reduced tochip size.

BACKGROUND

Bandpass filters (BPF) are commonly used in signal processing forvarious purposes. A BPF generally involves some form of resonator thatstores energy in a given frequency band. This resonator will have aninput coupling and an output coupling. Classical filters for electroniccircuit applications are built on this principle. The resonator may be,for example, in the form of a transmission line cavity, waveguidecavity, lumped inductor and capacitor components, or a crystal wheremechanical resonances of the crystal are coupled via electrodes to theelectrical circuit. The crystal could also be in a form of a small blockof ceramic material. An active form of a bandpass filter could includebuffers associated with the input and output resonator couplers suchthat the external coupling does not degrade the frequency selectivity ofthe resonator. Such an active filter is illustrated in FIG. 1, whichshows a generic active bandpass filter (BPF) having input and outputbuffers 102, resonator couplings 104, and a resonator 106.

The BPF of FIG. 1 can be made into a tunable BPF if the properties ofthe resonator can be adjusted. If they can be adjusted using passiveelements, then the BPF is a tunable BPF. The energy storage of theresonator can also be arranged with feedback in which signal from theoutput coupling is fed back into the input coupling. This is shown inFIG. 2, which depicts a generic BPF with a feedback path 110. Referringto FIG. 3, a gain block 112 and delay block 114 may be added thatcondition the feedback to modify the resonance slightly. The addition ofa gain block will turn a passive tunable BPF into an active tunable BPF.With this active feedback, more control is possible in which the phaseand the amplitude of the feedback can be controlled to give a narrowerbandwidth and finer control over the center frequency.

More specifically, the resonator feedback can be implemented in whichthe gain and the delay of the resonator feedback is assumed to beadjustable which modifies the frequency selectivity characteristics ofthe BPF. FIG. 3 shows control of the BPF feedback being implemented withthe delay block 114, where the adjustability of the circuit elements isdenoted by a diagonal arrow through the element.

If the overall loop gain (the loop consisting of the feedback path 110,couplers 104 and resonator 106) exceeds unity then the BPF becomes anoscillator, resonating at a frequency determined by the properties ofthe resonator 106 itself and the feedback loop 110. Backing off thefeedback gain such that the loop gain is slightly less than unityresults in a BPF with an arbitrarily narrow bandwidth. If the resonator106 selectivity is reduced such that it has a broader pass band then thefeedback can tune the filter over a broader range without becoming anoscillator.

Another general implementation is shown in FIG. 4 wherein the feedbackdelay element is replaced by a phase shifter 116, the phase shifterimplementing control of the feedback. Signal time delay and signal phaseshift are approximately analogous for narrow bandpass filters.

The circuit topology of FIG. 4 is essentially that of thesuper-regenerative amplifier filter that was developed back in the1930's (Armstrong). If the resonator 106 is based on a single inductorthen the feedback results in a Q-enhanced inductor circuit. If acapacitor is placed in parallel with the Q-enhanced inductor then atunable filter circuit results. Such circuits are published and wellknown.

The teachings in United States pre-grant publication no. 2013/0065542(Proudkii) entitled “Spectral Filtering Systems” are based generally onthe circuit of FIG. 4 with a fixed resonator element at low Q, oftenreferred to as a comb-line filter.

SUMMARY

There is provided a variable filter, comprising a signal loop definedbetween a signal input and a signal output, and a plurality of circuitelements connected in the signal loop. The plurality of circuit elementscomprises a frequency tunable resonator, and an adjustable scaling blockthat applies a gain factor that is adjustable in a range that comprisesa positive gain and a negative gain. There is also a controllerconnected to tune the frequency tunable resonator and to adjust the gainfactor of the adjustable scaling block between a negative gain factor toa positive gain factor.

According to further aspects, the variable filter may comprise one ormore of the following elements, alone or in combination. The frequencytunable resonator may comprise, but is not limited to, adjustableelements such as a varactor diode, variable dielectric capacitors,switched discrete capacitors, a MEMS variable capacitor, a fixedinductor, a variable inductor such as a MEMS variable inductor, or amechanically adjustable resonator. The plurality of circuit elements maycomprise a plurality of frequency tunable resonators. The plurality ofcircuit elements comprises two or more, or two or three frequencytunable resonators. One or more frequency tunable resonators may beconnected in a secondary signal loop that is connected within the signalloop, and each secondary signal loop may comprise a secondary adjustablescaling block. The adjustable scaling block may comprise a mainadjustable scaling block and is connected in series with each of thefrequency tunable resonators. The plurality of circuit elements maycomprise a plurality of adjustable scaling blocks. The controller may beconnected to independently tune two or more frequency tunableresonators. The controller may be programmed to selectively Q-spoil orQ-enhance one or more frequency tunable resonators. The variable filtermay further comprise a sensor that measures the frequency response ofthe signal loop, the sensor being in communication with the controller,wherein the controller is programmed to tune the one or more frequencytunable resonator(s), and control the gain factor of the one or moreadjustable scaling block(s) in response to the measured frequencyresponse to achieve a desired frequency response in the filter.

According to an aspect, there is provided a method of filtering asignal, comprising the steps of: providing a variable filter asdescribed above; and adjusting the filter by tuning the one or morefrequency tunable resonator(s) and adjusting the gain factor of eachadjustable scaling block to achieve a desired frequency response in thefilter.

According to further aspects, the method may include one or more of thefollowing steps, alone or in combination. Adjusting the filter maycomprise independently tuning two or more frequency tunable resonators.Adjusting the filter may comprise Q-spoiling or Q-enhancing at least onefrequency tunable resonator. The method may further comprise the step ofmeasuring the frequency response of the signal loop, and using thecontroller to tune the one or more frequency tunable resonators andadjust the gain factor of each adjustable scaling block in response tothe measured frequency to achieve a desired frequency response in thefilter.

According to an aspect, there is provided a programmable filter,comprising a plurality of variable filters as described above, and aswitch matrix connected to the inputs and the outputs of the pluralityof variable filters. The switch matrix is configurable to connect one ormore variable filters in more than one signal path configurations. Acontroller is connected to tune the frequency tunable resonators, adjustthe gain factor of the adjustable scaling blocks, and configure theswitch matrix between signal path configurations in order to achieve adesired frequency response in the filter.

According to a further aspect, the switch matrix may compriseconnections for selectively connecting one or more variable filters in asignal loop.

According to an aspect, there is provided a variable filter, comprising:a signal loop defined between a signal input and a signal output; aplurality of secondary signal loops connected in the signal loop, eachsecondary signal loop comprising a frequency tunable resonator and asecondary adjustable scaling block that applies a gain factor that isadjustable in a range that comprises a positive gain and a negativegain; a main adjustable scaling block; and a controller connected totune each of the frequency tunable resonators and to adjust the gainfactor of each of the main and secondary adjustable scaling blocks, thegain factors being adjustable in a range that comprises a negative gainfactor and a positive gain factor.

According to an aspect, there is provided a multiband filter circuit,comprising a plurality of filter elements, wherein the plurality offilter elements are selected from a group consisting of a variablefilter as described above. Two or more filter elements may be connectedin parallel or in series.

According to an aspect, there is provided a circuit comprising aplurality of resonators having a variable center frequency that has ameans of varying the center frequency of the filter passband, a scalingcircuit that can scale the amplitude output of the bandpass filter, afeedback path, an input coupler, and an output coupler wherein thescaling factor or gain of the scaling circuit can be adjusted andcontrolled.

According to other aspects, there may be a plurality of resonators, suchas two or three resonators, and the element to control the frequency maybe a variable capacitor, a variable dielectric capacitor, a variableinductor, a variable dimension of a resonator length, a MEMS device, orother known structure. The circuit is preferably a loop circuit, suchthat the various components may be in series in different orders. Thecircuit preferably allows for Q-spoiling of at least one resonator ofthe filter. The resonator may be a series LC, parallel LC, or a thirdorder bandpass filter. If there is more than one resonator, theresonators may be individually tuned such that the resonance frequenciesmay be staggered. Each resonator may have an individual scaling circuitthat affects the Q of the resonator individually. The scaling circuitsmay be a variable resistor, FET, or other known device that permits fora range of gain factors that includes both positive and negative gains.The resonators may be configured by incrementing or decrementing theresonator parameters based on an output characteristic of the overallfilter response of the circuit, such as by measuring the overall filterdominant pole location based on the measured or inferred impulseresponse. There may be a plurality of filters, switches, and connectionsin a configurable structure that may be configured in such a manner asto string several filters in series to realize a higher order filter,such as a Butterworth or Chebyshev bandpass filter. The filter may alsobe used as a band reject filter.

In other aspects, the features described above may be combined togetherin any reasonable combination as will be recognized by those skilled inthe art.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features will become more apparent from the followingdescription in which reference is made to the appended drawings, thedrawings are for the purpose of illustration only and are not intendedto be in any way limiting, wherein:

FIG. 1 is a block diagram of a prior art active bandpass filter.

FIG. 2 is a block diagram of a prior art active bandpass filter with afeedback loop.

FIG. 3 is a block diagram of a prior art bandpass filter with a delayelement in a feedback loop.

FIG. 4 is a block diagram of a prior art bandpass filter with a phaseshifter in the feedback loop.

FIG. 5 is a block diagram of a variable filter with a variable resonatorand a variable scaling block in a signal loop.

FIG. 6 is a graph showing the bandpass characteristics of a variablefilter.

FIG. 7 is a block diagram depicting a possible variant of the variablefilter.

FIG. 8 is a block diagram of a bandpass filter without a resonatorblock.

FIG. 9 is a graph of the periodic “comb-line” response of the bandpassfilter shown in FIG. 8.

FIG. 10 is a depiction on a complex plane of the first passband pole ofthe bandpass filter shown in FIG. 8.

FIG. 11 is a graph depicting the frequency response of the bandpassfilter of FIG. 8 in the vicinity of the first resonance band.

FIG. 12 is a graph of the excess power gain vs. Q of the bandpass filterof FIG. 8.

FIG. 13 is a visual definition of an elemental resonator, with S-planepoles.

FIG. 14 is a block diagram of a first order variable filter CAP-1.

FIG. 15 is a root locus of the first order variable filter of FIG. 14.

FIG. 16 is a Bode plot of the first order variable filter of FIG. 14.

FIG. 17 is a block diagram of a Q-spoiler based on a resonant block inparallel with a FET variable resistor.

FIG. 18 is a block diagram of a second order variable filter CAF-2.

FIG. 19 is a root locus of the second order variable filter of FIG. 18.

FIG. 20 is a Bode plot of the second order variable filter of FIG. 18.

FIG. 21 is a Hock diagram of a third order variable filter CAF-3.

FIG. 22 is a Bode plot of the third order variable filter of FIG. 21.

FIGS. 23 and 24 are root locus of the third order variable filter ofFIG. 21 with different values for Q.

FIGS. 25 and 26 are root locus of a fourth order variable filter withdifferent values for Q.

FIG. 27 is a simplified block diagram of an example of a first ordervariable filter.

FIG. 28 is a root locus of the variable filter depicted in FIG. 27 withQ-enhancement.

FIG. 29 is a root locus of the variable filter depicted in FIG. 27 withQ-spoiling.

FIG. 30 is a plot of the pass band of the variable filter depicted inFIG. 27 comparing Q-enhancement and Q-spoiling.

FIG. 31 is a plot showing the effect of varying the resonant frequencyon the pass band of the variable filter of FIG. 27.

FIG. 32 compares the frequency response of the first, second, and thirdorder variable filters.

FIG. 33 is an example of a Q-enhanced, Chebyshev bandpass filter.

FIG. 34 is a block diagram of three first order variable filtersconnected in series

FIG. 35 is a block diagram of three first order variable filtersconnected in series with an additional feedback path.

FIG. 36 are graphs depicting the effect of an additional feedback pathon the movement of the poles.

FIG. 37 is a root locus of three cascaded, first order variable filters.

FIG. 38 is a frequency response of the variable filter shown in FIG. 34for different values of G.

FIG. 39 is a frequency response of the variable filter shown in FIG. 35with neutral level two feedback.

FIG. 40 is a root locus for negative (left) and positive (right) leveltwo feedback of the variable filter shown in FIG. 35.

FIG. 41 is a plot of the frequency response of the variable filter shownin FIG. 35 with positive, neutral, and negative level two feedback.

FIG. 42 is a Bode plot of the variable filter shown in FIG. 35 withstaggered resonator frequencies.

FIG. 43 is a root locus of the variable filter shown in FIG. 35 withstaggered resonator frequencies.

FIG. 44 is a plot of a passband of a first order variable filter afterbeing subjected to a tuning process.

FIG. 45 is a plot of a passband of a third order variable filter afterbeing subjected to a different tuning process.

FIG. 46 is a zero degree root locus of a third order variable filterrepresenting quasi-orthogonal control.

FIG. 47 is a block diagram of three variable filters connected inseries.

FIG. 48 depicts a typical frequency response of the circuit shown inFIG. 47.

FIG. 49 is a block diagram of variable filters connected in parallel.

FIG. 50 depicts a typical frequency response of the circuit shown inFIG. 49.

FIG. 51 is an example of an arbitrary combination of variable filtersthat result in a tri-band filter.

FIG. 52 is a block diagram of a lattice array of multiple variablefilters with switch matrices allowing for arbitrary filter development.

FIG. 53 is the block diagram of FIG. 52, configured to provide a desiredsignal path.

FIGS. 54a and 54b depict the frequency response of a 4^(th) orderButterworth bandpass filter.

FIG. 55 is a block diagram of four variable filters in series.

FIG. 56 is a frequency response of the circuit shown in FIG. 55.

FIG. 57 is a simplified realization of a second order section circuittopography of a parallel resonator circuit.

FIG. 58 is a simplified realization of a second order section circuittopography of a series resonator circuit.

FIG. 59 is a simplified realization of a circuit topography of a thirdorder bandpass filter.

FIG. 60 is a block diagram showing a series of cascaded resonators.

FIG. 61 is a block diagram of a series of cascaded resonators with afeedback loop.

FIG. 62 is a block diagram of a generalized controllable analog filter.

FIG. 63 is a block diagram depicting the cascading of multiple CAF-nanalog filters, each with individual and separate controls.

FIG. 64 is a block diagram of a sensor wireless transceiver using acontrollable analog filter.

FIG. 65 is a block diagram of an alternate sensor wireless transceiverusing a controllable analog filter.

FIG. 66 is a block diagram of a wireless communication channel using acontrollable analog filter.

FIG. 67 is a block diagram of a baseband model of a wirelesscommunication channel.

FIG. 68 is a spectral diagram depicting adjacent channel noise due towireless interference.

FIGS. 69a and 69b are graphs that represent the response of an RRCfilter.

FIG. 70 is a graph of the power spectral density of additive channelnoise.

FIG. 71 depicts a pole/zero pattern of a third order, discrete timeButterworth low pass filter.

FIG. 72 shows an example of an eye diagram and the signal noise ratio ofa receiver.

FIG. 73 depicts an example of the signal noise ratio as a function ofthe passband of the CAF-3.

FIG. 74 depicts an example of the signal noise ratio as a function ofthe relative offset of the CAF-3.

FIG. 75 is a block diagram of a CAF-1 calibration and stabilizingcircuit.

FIG. 76 is a graph depicting a rapidly alternating Q control thatmeasures the pole position based on exponential rise and decay.

FIGS. 77 through 81 are block diagrams of examples of phase delayelements.

DETAILED DESCRIPTION

The presently described bandpass filter uses a resonator element ofvariable frequency that may be adjusted such that the resonator centerfrequency is coincident with the desired center frequency of the tunablefilter with feedback. In this way the resonator bandwidth can be narrowenough to avoid spurious passband responses but still be able to tuneover a broad range in frequency. In addition, the gain element, whichmay also be described herein as a scaling block, is adjustable toprovide positive gain, negative gain, or zero gain. As will be describedin more detail below, this allows the scaling block to act selectivelyas a Q-enhancer and a Q-spoiler. This filter architecture is referred toherein as a controllable analog filter (CAF). A block diagram of thesimplest form of the CAF with a variable resonator 502 and a variablescaling block 504 in a signal loop is given in FIG. 5. The descriptionbelow relates primarily to the design and control aspects of the CAF.The CAF is intended to be used in the transmitting, receiving, and/orprocessing of communications signals for various purposes, some of whichare described as examples below. Generally speaking, a communicationsignal is a modulated signal used to transmit information or data.Typically, this involves an analog carrier frequency that has beenencoded with an information signal, using known methods, such asfrequency modulation, amplitude modulation, or phase modulation.

The fundamental operating principle of the CAF, which offers control ofbandpass characteristics, is shown in FIG. 6, where the wide dashedtrace 602 is the resonator frequency response at an initial setting. Thenarrow dashed trace 604 is the sharper frequency response of the closedloop filter set for a narrower bandwidth at the initial frequencysetting. Assume that the resonator is now tuned upward in frequency tothe wide solid trace 606 as indicated by the black arrow. The narrowsolid trace 608 is the closed loop response that results at the newresonator response frequency.

The CAF may be of different orders, such as a first, second and thirdorder filter. The order of the CAF depends on the filter order of theresonator. The CAF may be designed with higher orders beyond the thirdorder, although the complexity of controlling higher order circuits willincrease. Some general comments on the filter order are given below.

-   -   A first order CAP (CAT-1) would include one resonator that has a        single dominant resonance pole pair.    -   A second order CAF (CAF-2) would include two such resonators        that have two dominant pole pairs, an example being a cascade of        two coupled LC tank resonators.    -   A third order CAP (CAF-3) would include three such resonators        having three dominant pole pairs.    -   Fourth and higher order CAFs are possible, however these may        result in spurious resonance bands when a scaling block feedback        is applied (as in FIG. 5), and generally result in a BPF with        more challenging tuning characteristics. This will be discussed        in more detail below.

The examples discussed herein will relate primarily to first, second andthird order CAFs, although it will be understood that the teachingsherein may also apply to higher order filters if desired.

In the various examples presented herein, the circuits are, forconvenience, typically depicted in the style of FIG. 5, which shows anarrangement having a main path 508 and a feedback path 510, andgenerally with the gain block 504 (which may also be referred to as ascaling block and which can have both positive or negative values) onthe feedback path 510. This is done for consistency and to make it easyto compare circuits. However, the circuit may be more appropriatelyconsidered as a loop with appropriate input and output couplings, wherethe loop is formed from what would otherwise be the main path 508 andthe feedback path 510, and the elements are connected in series withinthe loop. As a loop, the order of the components in the loop can bechanged without affecting the resonance characteristics of the CAF-n. Anexample of a circuit with the elements in a different arrangement shownin FIG. 7, where the gain block 504 is connected in the main path 508,and the variable resonator 502 is connected in the feedback path 510. Itwill be understood that other variations of the CAF circuit topologiesare also possible, and will depend on the number of elements used in therespective circuit.

Theoretical Development of a Bandpass Filter with No Resonator

Before presenting the theory of the CAF, the theoretical performance ofa CAF-0 (a CAF with no resonator) is discussed. This would be a basicbandpass filter of the type shown in FIG. 8, which has a gain block 802,attenuator 804 and variable phase shift or delay 806.

The phase shift 806, either through a transport delay or a phaseshifting element, varies with frequency, imparting to the feedback loopits frequency selectivity. Based on this circuit description, the lineartransfer function of the depicted circuit can be stated as a Laplacetransform (frequency domain response) written as

${H_{{ABF}\; 0}(s)} = \frac{G}{1 - {AGe}^{- {sP}}}$

where P is the phase shift. We can write this in terms of the physicalfrequency f with the mapping s→j2π f as

${H_{{ABF}\; 0}(f)} = \frac{G}{1 - {AGe}^{{- j}\; 2\;\pi\;{Pf}}}$Note that this is periodic in frequency with a period of f_(p)=P⁻¹. Thatis, although the CAF-0 has no specific resonator element, resonancebands occur whenevermod(f _(p) ,P ⁻¹)=0.In the following numerical example, assume that the delay P is 100 psecand that G=1 and A=0.9. Hence resonance will occur at 0 Hz, 10 GHz, 2.0GHz, etc. Another observation is that all the resonant bands have thesame in-band gain of 20 dB which is a result of

$\begin{matrix}{{{H(f)}}_{f = \frac{n}{P}} = \frac{G}{1 - {AGe}^{{- j}\; 2\;\pi\; P\frac{n}{P}}}} \\{= \frac{G}{1 - {AGe}^{{- j}\; 2\;\pi\; n}}} \\{= \frac{G}{1 - {AG}}} \\{= {\frac{1}{1 - {.9}} = \left. 10\Rightarrow{20\;{dB}} \right.}}\end{matrix}$

Such filter performance is commonly referred to as a comb-line filter inreference to the transfer function plotted in FIG. 9.

Considering the fundamental resonance at f=1/P, we can see by changing Pwe can get any arbitrary resonance. The response at DC and the higherorder harmonics can easily be attenuated by a suitable low pass and highpass filter that are cascaded in series with the circuit of FIG. 8.

The Q and damping factor D of the CAF-0 are derived from the Laplacetransform of the CAF-0 given before as

${H_{{ABF}\; 0}(s)} = \frac{G}{1 - {AGe}^{- {sP}}}$

It is enough to determine the principle solution for the pole at zerofrequency on the real axis given as

${AGe}^{- {sP}} = {{1 - {sP}} = {\ln\left( \frac{1}{AG} \right)}}$$s = {\sigma = {\frac{1}{P}{\ln({AG})}}}$

Now consider the dominant passband pole shown in FIG. 10.

We have

${\omega_{o}D} = {\sigma = {{- \frac{1}{P}}{\ln({AG})}}}$

where D is defined as the damping coefficient and ω₀ is the naturalresonant frequency in rads/sec. This gives

$\begin{matrix}{D = {{- \frac{1}{\omega_{o}}}\frac{1}{P}{\ln({AG})}}} \\{= {{- \frac{P}{2\;\pi}}\frac{1}{P}{\ln({AG})}}} \\{= {{- \frac{1}{2\;\pi}}{\ln({AG})}}} \\{= {{\frac{1}{2\;\pi}{\ln\left( {1 + {AG} - 1} \right)}} \approx \frac{1 - {AG}}{2\;\pi}}}\end{matrix}$This Taylor expansion simplification is based on the assumption thatAG≈1. The approximate relation to Q (valid for larger Q say Q>10) isgiven as

$Q = {\frac{\omega_{o}}{2\;\sigma} = {\frac{\omega_{o}}{2\;\omega_{o}D} = {\frac{1}{2\; D} = \frac{\pi}{1 - {AG}}}}}$This makes sense in that Q increases to infinity as the loop gain AGbecomes 1. FIG. 11 shows an example of the calculation of the responsearound the first resonance pole for AG=0.90 The portion represented by abroken line is the response within the 3 dB bandwidth from which theexpression of Q can be validated.

A more direct way of calculating the Q factor is to have

$Q = \frac{\omega_{o}}{2\;\Delta\;\omega}$

where Δω is determined from the 3 dB point of H(s). At the real axispole

1 = AGe^(−j(ω_(o) + Δ ω)P)$\frac{1}{AG} = {{e^{{- j}\;\Delta\;\omega\; P}} = 1}$${1 - \frac{1}{AG}} \approx {\Delta\;\omega\; P}$$\frac{{AG} - 1}{AG} = {{\Delta\;\omega\; P} = \frac{2\;\Delta\;\omega\;\pi}{\omega_{o}}}$From this we have

$Q = {\frac{\pi\;{AG}}{1 - {AG}} \approx \frac{\pi}{1 - {AG}}}$

which was obtained before.

Next consider the power gain that is associated with a given Q. Thepower gain is given as

$G_{p} = {\left( \frac{G}{1 - {AG}} \right)^{2} = {{\frac{G^{2}}{\pi^{2}}\left( \frac{\pi}{1 - {AG}} \right)^{2}} \approx {\frac{G^{2}}{\pi^{2}}Q^{2}}}}$The excess gain resulting from the feedback path, denoted as G_(ex), isthe power gain normalized by the open loop gain (no feedback path) as

$G_{ex} \approx {20\;{\log\left( \frac{Q}{\pi} \right)}}$

in dB. This is plotted in FIG. 12.

Elements of the CAF

Before Turning to Structure of the CAF, the Following CAF Elements areDiscussed.

Tunable CAF Resonator (R)

The resonator block 1301 is denoted as R in FIG. 13. R is represented bytwo poles in the S-plane as is also shown in the diagram on the right ofFIG. 13. The two poles are represented by ‘x’. By incorporating afeedback loop across the resonator as in FIG. 14, Q control is madepossible. This is discussed in detail later. As depicted, resonatorblock 1401 with feedback incorporating gain or scaling block 802 asshown in FIG. 14 is the basic resonator element that has two controlinputs 1302 and 1304 one for changing the frequency (1302), and theother for changing the Q (1304).

Depending on the circuit being implemented, the resonator can beimplemented either as a resonator element, such as a LC tank circuit, oras a Second Order Section (SOS) filter element. The SOS implementationwill be discussed in more detail below.

As will be understood from the discussion herein, there are a number ofpossible combinations of resonators and scaling blocks, and signal pathsthat can be used in designing a CAR The particular design will depend onthe desired circuit performance, as is discussed elsewhere. Generallyspeaking, the CAF will include a feedback loop comprising a desirednumber of resonators and a scaling block. Each CAF may be connected inparallel or in series with other CAF elements, or other circuitelements, and may have an additional level two feedback loop thatcomprises multiple CAR. In addition, there may be nested loops within aCAF element, comprising a loop for each resonator, or subset ofresonators. As used herein, the term “level two feedback” is intended torefer to a feedback or circuit loop that provides a feedback path aroundmultiple CAF-n elements in series. This may also include what couldotherwise be referred to as a level three or level four feedback.

The resonant frequency of R can be varied with some component includedin the resonator circuit. Typically, this may be accomplished using avaractor diode, or a variable dielectric capacitor may be used for avariable capacitance, in which case the ‘f control’ in FIG. 13 would bean analog bias voltage. Other variants that allow the resonant frequencyto be varied may also be used, such as a discrete capacitance that isswitched in or out of the circuit and hence ‘f control’ can be a digitalsignal. Another variant is that a MEMS variable capacitor or a MEMSvariable inductor could be used where ‘f control’ is a bias controlvoltage or current signal applied to the MEMS device. The variablecapacitance or inductance can also be realized by mechanical tuning of acomponent. For instance, R could be a microwave resonance cavity inwhich one or more dimensions of the cavity are mechanically adjustableby some mechanism supplying ‘f control’.

The two poles of R are a conjugate pair and cannot be controlledindividually. Hence to simplify the description we consider only thepositive frequency pole. We therefore consider the elemental resonatoras having a single pole in the domain of s (that is the region of s withpositive imaginary component). R is a two port device with a transferfunction given in the Laplace domain, denoted as above as a standardsecond order bandpass transfer function H_(R)(s):

${H_{R}(s)} = \frac{as}{s^{2} + {2\; D\;\omega_{n}s} + \omega_{n}^{2}}$Q Control Scaling Block

The ‘Q control’ 1304 in FIG. 13 above can comprise a control deviceassociated with the resonator that controls the component Q of thecapacitance or the inductance or resonant cavity. If the Q controlincreases the component Q, this is referred to herein as Q-enhancement.If the Q control decreases the component Q of the resonant cavity, thisis referred to herein as Q-spoiling. Q-enhancement is equivalent todecreasing D which moves the resonant pole of R closer to the jω axis ofthe S-plane. Q-spoiling moves the resonant pole of R further from the jωaxis hence increasing D. It has been found that Q-enhancement andQ-spoiling may be used selectively to move a resonant pole towards oraway from the jω axis to synthesize an arbitrary multi-pole filterfunction (plurality of R's).

Scaling blocks 802, as in FIG. 14, are provided in order to enablebetter control over the feedback response. The gain factor for eachscaling block 802 is variable and comprises a gain that includes bothpositive and negative gain values. For example, if the gain of thescaling block 802 is greater than zero, there results Q-enhancement. Ifthe gain of the scaling block 802 is less than zero, there resultsQ-spoiling.

In general, there will be an additional level two scaling block for eachloop or secondary loop in a CAF-n element as discussed below. As anexample, for a CAF-3 circuit element (see FIG. 35 for reference), wherea series of three CAF-1 elements are connected within a loop and can beseparately controlled, there may be four scaling blocks—a scaling block802 surrounding each resonator element 1401 in loops 110, and one leveltwo feedback scaling block 802 a in loop 110 a, as will be discussedbelow.

Generally, each scaling block will be capable of enabling Q-enhancementresonators and Q-spoiling resonators independently. Alternatively, theresonator may be a Q-enhanced resonator, which uses an amplifier thatonly allows for Q-enhancement. As noted above, the Q-enhanced resonatorwould still be nested within the feedback loop of the CAF-n elementcomprising a scaling block to override the Q-enhancement and provide adesired Q-spoiled performance as required. This will, of course, beapparent from the fact that the resonator may be any type of frequencytunable resonator comprising, but not limited to, a varactor diode, aswitched discrete capacitor, a variable dielectric capacitor, a variablecapacitor, such as a MEMS variable capacitor, a fixed inductor, avariable inductor, such as a MEMS variable inductor, or a mechanicallyadjustable resonator.

Topology of the CAF-1

There will now be described a first order of the CAF circuit, denotedCAF-1, which comprises a single resonator component 1401, a single gainor scaling block 802, and a combiner 1404 for closing the feedback loopas depicted in FIG. 14. This can be described in a simplified way if thecenter frequency control of the CAF-1 is omitted. This provides anintuitive method of understanding the CAF-n variants. In one example,resonator 1301 may be a second order bandpass filter with a transferfunction of:

$\frac{1}{s^{2} + {2\; D\;\omega_{o}s} + \omega_{o}^{2}}$

with coefficients evaluated based on D and ω_(o). The gain G 802 isvariable and controls the closed loop Q. Note that at resonance thephase shift through the resonator 1401 is ideally 0 degrees. In thephysical implementation the phase shift will not be zero in general dueto parasitics and transport effects, but these can be ignored in thisevaluation: the implemented circuit will have a phase shifter associatedwith G 802 that will compensate for any parasitic and transport phaseeffects. To vary the frequency it is necessary to change ω_(o) of theresonator in the CAF-1, but this is ignored in this section.

It should be noted that, according to the notation used herein, thefirst order CAF-1 has a resonator of second order. What is referred toin “order” is the number of Second Order Sections (SOS) used that makeup the overall resonator. An SOS transfer function refers to a Laplacefunction of frequency variables that are second order in thedenominator. In the present context the SOS, as seen above, will alwayshave the form of

${H_{SOS}(s)} = \frac{as}{s^{2} + {2\; D\;\omega_{o}s} + \omega_{o}^{2}}$

where ω_(o) is the resonance frequency in radians per second, D is thedamping coefficient, and a is a real constant. The mapping to fin FIG.13 isf _(n)=ω_(n)/2πThe mapping to Q is given by the conventional definition of

$Q = {\frac{{center}\mspace{14mu}{frequency}}{3\;{dB}\mspace{14mu}{bandwidth}} = {\frac{\omega_{n}}{2\; D\;\omega_{n}} = \frac{1}{2\; D}}}$In this discussion, {f_(n),Q} may then be used interchangeably with{ω_(n),D}.

An insightful analysis of the operation of the CAF-1 is possible withthe use of the root locus method. The root locus is a standard method ofdetermining the poles of a closed loop system given a variable loopgain. The outcome of the root locus calculation in the present contextis the trajectory of these closed loop poles as they change withvariations in the loop gain G as shown in FIG. 15 for example. With thiswe can get an understanding of any spurious passbands and any tuninglimitations of the CAF-1. As an initial illustrative example let usassume that ω_(o)=1 and D=0.5, in which the CAF-1 SOS resonator has avery low Q for this example. The Bode plot of the transfer function ofthe resonator is shown in FIG. 16. Note that the phase change withfrequency is rather gradual around resonance due to the high dampingfactor (low Q) assumed in this example.

Now we consider the effect of the feedback gain G on the closed looppoles. This is calculated by the 0 degree root locus calculation and isshown in FIG. 15. Here the poles of the CAF-1 SOS resonator arerepresented by an x (1502 or 1504). Line 1506 is the closed loop roottrajectory as the closed loop gain G is increased from 0 to 1.2. This isthe trajectory for the pole indicated by number 1502. The root locus ofthe conjugate closed loop root trajectory 1504 is line 1508. Note thatthese move towards the jω axis of the S-plane (root locus in the domainof the complex frequency variable s=π+jω) indicating a progressivelyhigher Q as the closed loop gain U is increased. If the closed loop gainwas decreased as with Q spoiling, then the root locus of the poles 1506and 1508 would move away from the jω axis (not shown in FIG. 15). InFIG. 15, when the root locus trajectory crosses into the right handplane the closed loop roots are unstable.

In this unstable region of operation the CAF-1 is not usable and roottrajectories cease to be meaningful. Hence we only need to plot over therange of G in which the closed loop poles remain in the left hand plane(LHP). Incidentally, for the value of G for which the closed loop polescoincide with the jω axis, the CAF-1 oscillates at the resonantfrequency of ω_(o), which is normalized in this example to ω₀=1. Theradial dotted lines in the root graph indicate the damping value of D. Qcan be related to D based on the relation of Q=½D. Also in this example,the gain G where the root trajectories cross the jω axis and the CAF-1becomes unstable is S=1.

As the Q of the SOS CAF-1 resonator is decreased (note: this is not theclosed loop system Q), the filtering in the initial forward path islimited by the 20 dB per decade change in the frequency. A problem withthis is that the out of band signals and broadband noise is notsignificantly attenuated by the first forward pass through of thesignal. As the CAF-1 Q increases, these out of band signals areeliminated in the output only if they are subtracted at the summingblock 1404 in the CAF-1 circuit of FIG. 14. This implies that the signalflowing through the gain block 802 has to be large. To reduce this, onehas the option of 1) raising the Q of the SOS feedback resonator, or 2)adding an additional SOS feedback resonator. Adding an additional SOSfeedback resonator results in a CAF-2 which is described below.

Another way of implementing a variable Q for the SOS resonator is the‘Q-spoiler’ which is implemented by a variable resistive element in theSOS. This affects the damping coefficient of the SOS which could havebeen designed to have a higher Q than typically desired. The variableresistor reduces (spoils) the Q such that the poles of the SOS arefurther from the jω axis into the LHP as mentioned above. This is adegree of freedom (DOF) that allows for higher attenuation of outliersthan if an SOS with a fixed lower Q was implemented. One embodiment ofthe Q-spoiler circuit is shown in FIG. 17 based on a parallel resonanceSOS 1401. In this case the Q-spoiler is implemented with a PET 1702operating in the triode region in parallel with a resonator 1401 andcontrolled by a Q-spoiler control voltage 1704 to provide an equivalentvariable resistor function. In another implementation the FET 1702 couldbe implemented with a PIN diode. It will be understood that these designoptions may be incorporated into any of the variable filter circuitsdescribed herein.

Topology of the CAF-2

A topology of the CAF-2, comprising two SOS resonators 1301, is shown inFIG. 18. The unit gain buffer 102 separating the resonators is forimplementation purposes when necessary to isolate the resonators fromeach other. Again combiner 1404 provides feedback loop closure. Thetransfer functions of each resonator 1310 are:

$\frac{1}{s^{2} + {2\; D\;\omega_{o}s} + \omega_{o}^{2}}$For purposes of example, the parameter values for ω₀ and D are selectedas ω_(o)1 and D=0.5. The zero degree root locus for the CAF-2 is givenin FIG. 19.

The open loop roots are indicated by ‘x’, with two poles located at 1902and two conjugate poles at 1904 as required for this dual SOS resonatorconfiguration. By definition, the “dominant” pole is always closest tothe jω axis, while the “secondary” pole is furthest from the jω axis. Wesee the dominant root trajectories 1906 b and 1908 b move towards the jωaxis as the closed loop gain G is increased, while the other set oftrajectories 1906 a and 1908 a move away from the jω axis. Hence the tworesonator CAF-2 will still have a dominant pole pair that behaves as thesingle resonator CAF-1. At a loop gain G=1, the root trajectories 1906 band 1908 b cross the jω axis into the right hand plane (RHP), the Q ofthe CAF-2 becomes infinite, and the overall circuit becomes unstable. Aswith the CAF-1, stability is only possible when the root locus stays inthe DIP which can only occur for closed loop gain G<1 for eachindividual resonator.

The advantage of the two resonator CAF-2 is that the attenuation of outof band signals is larger and therefore the interference signal is less.The poles moving along 1906 a and 1908 a into the left hand plane awayfrom the jω axis still contribute to attenuation of the out of bandspectral components. Furthermore, as out of band spectral components arebetter filtered by the CAF-2 double SOS resonator, these out of bandcomponents flowing around the feedback loop will be less. This isimportant because the intermodulation distortion performance of the loopcomponents does not have to be as high as in the case of the CAF-1.

Another possible trade off is that the same out of band rejection withtwo SOS resonators with a smaller closed loop Q is achieved with asingle SOS resonator with a larger closed loop Q. This is important asthe power gain is proportional to the square of Q as shown above withrespect to the CAP-0, and which is approximately valid for the CAF-nmore generally. For large Q, the large power gain can become a practicalimplementation limitation. The Bode plot for the double resonator CAP-2is given in FIG. 20. Note that the CAF-2 transition steepness is 40 dBper decade change in frequency, whereas for the single CAF-1 SOSresonator it is 20 dB per decade change in frequency.

Topology of the CAF-3.

Turning to the third order CAF-3, FIG. 21 shows a topology of the CAF-3which has three cascaded SOS resonators, each of which includes afeedback loop, a method for changing the center frequency of theresonator, and a method for changing the Q of the resonator. As with theCAF-2, unit gain buffers 102 are placed between all of the resonators1301 for isolation, and a combiner 1404 to close the feedback loop. Itis important to note the ability to individually control both the centerfrequency and gain of the individual resonators in this and other CAF-nconfigurations. Initially, we shall set the center frequency of eachresonator to be the same, and will discuss the CAF-3 with differentcenter frequencies later.

The Bode plot of the triple resonator, each with the same centerfrequency, is shown in FIG. 22 where the out of band open loopattenuation of the triple resonator is seen to be 60 dB per decade infrequency which is of significance as it is based on low Q resonators.The zero degree root locus is shown in FIG. 23 for a D=0.5 or a Q=1. Theroot locus is interesting in that there are three root trajectories 2306a/b/c and 2308 a/b/c emanating from each triple of open loop poles 2302and 2304 marked again by the ‘x’, although image scaling makes the threeindividual roots impossible to differentiate. Note that one of the roottrajectories 2306 a/2308 a follows the ω_(n)=1 contour exactly asbefore, while the other root 2306 b/2308 b goes further into theleft-hand plane (LHP) and does not influence the circuit. However, thethird pole trajectories 2306 c/2308 c start to move toward the jω axis.This potentially gives rise to a spurious mode that is at much lowerfrequency than the intended passband. However, at the gain G where thedominant pole gets sufficiently close to the jω axis to realize thedesired higher Q closed loop poles, this potentially troublesome pole isstill far from the jω axis and causes a negligible spurious response ina practical implementation. Serendipitously, as the Q of the CAF-3 SOSresonators are increased such that D decreases, this potentiallytroublesome root goes further into the LHP as shown in the zero degreeroot locus example of FIG. 24 which is calculated for a D=0.1, or aresonator Q=5, with trajectories 2406 c from pole 2402 and 2408 c frompole 2404. As with all CAF-n implementations, stability is achieved whenthe root locus stays in the LHP, which occurs for closed loop gain G<1for each individual resonator.

Topology of the CAF-4

For the sake of completion, a short description of the CAF-4 is alsogiven. This is of higher complexity than the CAF-3 and has spuriousresponses that could be detrimental in some applications. The zerodegree root locus for the CAF-4 with D=0.5, or a resonator Q=1, is givenin FIG. 25, with trajectories 2506 a/b/c/d from pole 2502 andtrajectories 2508 a/b/c/d from pole 2504. The zero degree root locus forthe CAF-4 with D=0.1, or a resonator Q=5, is given in FIG. 26, fortrajectories 2406 b from pole 2402 and trajectories 2608 a/b/c/d frompole 2604. Note that in FIG. 25 where the Q of the SOS is very low, thespurious passband resulting from the 2506 d or 2508 d paths can beacceptable. As in the third order CAF-3 case, the CAF-4 root leading tothe spurious frequency response is still far from the jω axis andcorresponds to a much lower frequency which can be suppressed with a lowpass filter. However, considering the root locus in FIG. 26corresponding to the higher resonator Q, there are two root trajectories2606 d and 2608 d that do not really move further into the LHP andconsequently create the spurious frequency responses Which are generallyundesirable in filtering applications.

In summary, the CAF-1 can give good band pass filtering performance formany applications. However, the CAF-2 and CAF-3 filters can give moreflexibility for tailoring to an application. The CAF-3 will provide thebest rejection of the out of band signals for typically encounteredclosed loop Q values. It is the configurable root trajectory of thesecond and third order CAF closed loop poles that is a key attribute ofthis innovation.

CAF-1 Detailed Example

In this section an example will be provided of a CAF-1 filter circuitwhich consists of feedback around a single pole pair. An applicationcircuit could be as shown in FIG. 27. Here the single resonator 2702 isa fixed resonator circuit with a feedback gain 2704. The gain G of block2704 can be negative for Q-spoiling or positive for Q-enhancement. It isunderstood that while gain block 2704 is shown as a two port gain blockthat it can be arranged as a one port gain block with either negative orpositive resistance. Negative resistance would result in G beingequivalently greater than zero and provide Q-enhancement. Positiveresistance, on the other hand, is equivalent to a negative G providingQ-spoiling.

The root locus of the positive frequency closed loop pole for positive Gis shown in FIG. 28. This corresponds to the Q-enhancement case wherethe close loop pole moves towards the jω axis. Likewise the root locusfor negative G is shown in FIG. 29. This corresponds to the Q-spoilingwhere the close loop pole moves away from the jω axis.

FIG. 30 shows an example of the passband response with neutral Q (G=0),Q-enhancement (G=0.15) and Q-spoiling (G=−0.5). Note how the bandwidthis easily modulated with a small change in the feedback gain G.

Consider the case where the resonator R comprises a means to vary theresonance frequency of the CAF-1. A frequency response example is givenin FIG. 31 in which the resonance of R has normalized frequency values ωof 0.9, 1 and 1.1 for G=0.15. The time required to tune from onefrequency to the next is approximately equal to the reciprocal of thebandwidth of the CAF-1.

Comparison of CAF-1, CAF-2, and CAF-3.

In this example, the operation of the CAF-1, CAF-2, and CAF-3 arecompared. The resonators in the three feedback filters are the same withD=0.1 and a normalized resonance of ω=1. The Q-enhancement is tuned inthe three filters separately such that they have approximately the sameclose in pass-band response. Values are G=0.13 for the CAF-1, G=0.07 ofCAF-2 and G=0.002 for CAF-3. The pass band frequency responses areplotted in FIG. 32. As noted, the benefit of the CAF-3 is the higherrejection of the frequency components that are farther from the centerfrequency as compared to the CAF-2. Also the CAF-2 has better frequencyselectivity in comparison to the CAF-1 as expected.

Applications of the CAF-3 for Simplified Bandwidth Control

When connected in series, three CAF-1s can realize three resonant poles.This can be used to provide similar results as a 3^(rd) order Chebyshevtype bandpass filter, an example of which is shown in FIG. 33.

Referring to FIG. 34, the equivalent scheme with the three CAF-1 sshown, where the resonators 1401 each have a feedback path 110 with ascaling block 802 and are separated by buffers 102. In this example, thepoles of the three CAF-1s are generated with Q-enhanced inductors thatcan be set arbitrarily close to the jω axis.

In addition, referring to FIG. 35, a level two feedback path 110 a maybe wrapped around the three CAF-1 modules, such that the circuit willthen behave like a CAF-3.

What the additional CAF-3 level two feedback loop 110 a does is modifythe pole movement in the S-plane as illustrated in FIG. 36. The arrows3602 are for negative feedback (Q-spoiling) and the arrows 3604 are forpositive feedback (Q-enhancement). Note how the movement is differentfor the two cases.

If arbitrary placement of the poles to realize a certain filter responseis desired, then it is possible to provide a Q-enhance/spoil for eachindividual CAF-1 of FIG. 34. However, the control becomes more complexas six controls are necessary. Also, there is redundancy in the controlas the order of the resonators is generally irrelevant. This addsconfusion to the pole placement stability tracking algorithm. A simplercontrol is that of having a level two feedback loop 110 a as shown inFIG. 35. In that case, the feedback around each CAF-1 is driven from acommon control source (not shown), and each feedback loop has a gainblock (not shown), as described herein. Additionally, the outer controlloop 110 a is around the three individual CAF-1 resonators and also hasa gain block (not shown). Hence the first control for the CAF-1's movesthe three poles in unison towards or away from the pro axis. The CAF-3level two control can spread the outer flanking poles and cause thecenter pole to retreat slightly. This enables controlling the bandwidthof the filter while maintaining a similar transition rate.

For this CAF-3 implementation evaluation, there will be considered threeCAF-1 resonators with the following attributes:

Normalized Damping resonance frequency factor Resonator (f) (D) 1 1 .4 2.95 .42 3 1.05 .38This locates the pole at.s=2πf D+j2πf√{square root over (1−D ²)}Now consider that each, of these three resonators are with feedbackloops such that there are 3 cascaded CAF-1 modules. The root locus isshown in FIG. 37.

The ‘x’ 3702 a/b/c designate the positions of the poles with feedbackgain of 0. The gain is positive for right excursions 3704 a/b/c towardsthe jω axis (Q-enhancement) and negative for excursions 3706 a/b/c tothe left (Q-spoiling). In terms of negative resistance amplification(Q-enhancement), it would imply that the resistance is zero at theposition of the ‘x’, with positive resistance (Q spoiling) for leftexcursions of the root trajectory and negative resistance for rightexcursions. Note how the control goes along a contour of constantnatural resonance frequency. The range of the feedback gain for eachroot trajectory is −1<G<0.9.

FIG. 38 shows the frequency response of the three CAF-1s when thefeedback gain is 0, −1.0 and 0.9 showing the effect of Q-enhancement(positive G) and Q-spoiling (negative G) compared to neutral gain (G=0).Note that for this plot the peak amplitude has been normalized to 1 tomake the plot clearer.

Next consider a modified CAF-3 with three resonators in which theresonators are CAF-1's that have been Q-spoiled with a gain of G=−0.9.FIG. 39 shows the frequency response when such a CAF-3 when level twofeedback gain is G=0.

Next consider how we can make this look like a second order Chebyshevbandpass filter response by changing the CAF-3 level two feedback.

Referring to FIG. 40, the left plot is for the negative CAF-3 level twofeedback and the right plot is for the positive CAF-3 level twofeedback. Note how we can use this to adjust the position of theflanking poles relative to the center pole. FIG. 41 shows how the CAF-3level two feedback can be used to control the bandwidth of the filter.Positive level two feedback narrows the filter bandwidth and negativelevel two feedback broadens it. Only a very small amount of CAF-3 leveltwo feedback is needed for this control. In FIG. 41, the level twofeedback was 0, −0.002, and +0.002, as indicated.

As can be seen, the CAF-3 level two feedback control of FIG. 35 allowsfor an effective means of bandwidth control that can be practicallyimplemented.

Considered now a modification to the CAP-3 shown in FIG. 35, in whichthe SOS resonator 1401 pole locations are staggered in frequency andconsequently not collocated in the S-plane, as was considered earlier.This can provide more flexibility in the control of the CAF-3 closedloop frequency response. The advantage of this increased flexibility isthat different frequency responses can be achieved. This additionalflexibility is of significance in some applications which are beyond thescope of this disclosure. As an example of staggered frequencyresonators, consider the case of normalized SOS resonator naturalfrequencies of ω=0.9 rad/sec, 1.0 rads/sec and 1.1 rads/sec, with Dfixed at 0.15 (Q=3.33) for all three resonators. The Bode plot of thesethree SOS resonators cascaded is given in FIG. 42 indicating thepotential of a more flattened pass band, which is worth investigating.FIG. 43 shows the zero degree root locus of this configuration, which isvery similar to that shown in FIG. 23 where each of the three SOSresonators has the same center frequency.

Controlling the CAF Performance

The various elements in the CAF-n circuits may be controlled using acontroller. It will be understood that various types of controllers maybe used as is known in the art, comprising controller circuits andvarious microprocessors. Furthermore, while there is preferably a singlecontroller that controls the various elements of the CAF-n, there mayalso be multiple controllers, or various layers of controllers. Thecontrollers may be programmed to adjust the variables in the CAF-naccording to an algorithm, a lookup table, software, or according toother known strategies, all of which may depend on specificimplementation objectives and appropriate tradeoffs for thatimplementation. The controller may be programmed to respond to inputsfrom a user or from other circuit elements. In some circumstances, itmay be beneficial to measure the frequency response, such as an impulseresponse of the CAF-n using sensors or detectors downstream of theCAF-n. Using this approach, the CAF-n may be controlled by providing thecontroller with a desired frequency response, which then controls thevariables in order to achieve the desired frequency response. This mayallow for an iterative approach to be used, or for fine adjustments tobe made after the controller has approximated the desired frequencyresponse.

The control of the CAF-n is relatively simple in that the closed loopcenter frequency control and the closed loop Q are almost independentand the control optimization is strictly convex. In other words, thecontrol of the closed loop Q and center frequency can be doneindependently for many adjustments, which simplifies the system controlalgorithm.

In the following, an example of a CAF-n tuning and tracking scheme willbe shown. This is an example of an embodiment of such a scheme, and itis understood that a wide variety of such tuning and tracking algorithmscan be implemented by one skilled in the art. Consider a simple tuningscheme in which there exists a means of estimating the dominant polelocation of the closed loop CAF-n. This may be done, for example, bymeasuring the impulse response of the CAF-n and determining the resonantfrequency of ω_(o) and the damping coefficient D_(o). Alternately, thiscan be input as a user design target. Then a tracking loop determines 1)the overall feedback gain G, and 2) the SOS resonator(s) naturalfrequency denoted by ω_(r). In an embodiment of the CAF-n, G can bedetermined by setting a control voltage on the feedback amplifier, andω_(r) is set by adjusting the voltage on a varactor diode of the SOSresonant tank. Let D_(d) and ω_(d) be the desired damping and resonantfrequency respectively provided by the user design targets. The trackingloop is straightforward: if D_(o)<D_(d) then G is increasedincrementally. If ω_(o)>ω_(d) then ω_(r) is increased incrementally. Theloop iterates, updating G and ω_(r) sequentially indefinitely until thedesired response is achieved. In this way the initial configuration ofthe CAF-n is accomplished and also the temperature variations andcomponent aging of the CAF-n are ameliorated. Furthermore, the desiredD_(d) and ω_(d) can vary with time and the CAF-n will track these.

By way of example, consider a CAF-1 filter with D_(d)=0.02 and ω_(d)−1.We assume that the SOS resonator in this example has a damping ofD_(r)=0.4 and an initial resonance frequency of ω_(j)=0.9 The initialloop gain was G=0.4. These values are rather arbitrary with the soleintent of presenting a practical example.

The resulting frequency response of the CAF-1 after tuning is completeis shown in FIG. 44. Note the desired parameters of D_(d) and ω_(d) aremet.

As a second example, consider a CAF-3 filter with the same D_(d)=0.02and ω_(d)=1 requirements given. Again the SOS resonators in this examplehave initial damping of D_(r)=0.4 and initial resonance frequencies ofω_(r)=0.9. The initial loop gain is again G=0.4.

The resulting frequency response of the CAF-3 after control cycling iscomplete is shown in FIG. 45. Note the desired parameters of D_(d) andω_(d) are met.

The reason the CAF-1 through CAF-3 tracking is simple and robust is thatthe zero degree root locus for the dominant pole essentially follows thecircular curve of constant ω_(o) towards the jω axis. Movement in Gbasically moves ω₀ along this arc, and change in ω_(r) makes it moveradially. These motions are quasi-orthogonal. This is shown in FIG. 46for the CAF-3.

Parallel and Series Combinations, CAF-n Elements

Referring to FIG. 47 through 53, multiple CAF-n elements, each of whichmay include one or more loop circuits with resonators and gain elementsas discussed above, can be configured in series and parallelcombinations to realize multi-pole bandpass filters. As an example,referring to FIG. 47, three CAF-1 elements 4701, 4702, and 4703 areshown to be connected in series. In this example, each CAF-14701/4702/4703 has a frequency response of that of a first orderbandpass filter (one resonant pole in the positive frequency region) andeach CAF-1 is adjusted such that it has the appropriate Q and resonantfrequency commensurate with the desired overall filter response. Forexample, the transfer function of CAF-1 4701 is set at:

$\frac{s}{s^{2} + {{.1}\; s} + 0.96}$

the transfer function of CAF-1 4702 is set at:

$\frac{s}{s^{2} + {{.1}\; s} + 1}$

and the transfer function of CAF-1 4703 is set at:

$\frac{s}{s^{2} + {{.1}\; s} + 1.02}$A typical response of three CAF-1 elements 4701/4702/4703 in series isshown in FIG. 48. The series connections of CAF-1's are useful forrealizing single band bandpass filters. To implement multiple bands, aparallel connection of CAF-n's may be used. An example of a filtertopology providing two separate bands is given in FIG. 49. This isaccomplished by a parallel connection of two CAF-3 filters 4901 and 4902centered at the respective bands. The general response of such a filteris shown in FIG. 50.

In all of the above parallel and serial topologies involving multipleCAF-n elements, the level two feedback of FIG. 35 may be considered aswell.

The bandpass filters discussed above are examples with practicalutility. It will be understood that other series and parallelcombinations of CAF-n's can be used to realize bandpass filters designedto achieve a desired frequency response.

Arbitrary Series and Parallel of CAF-n's

Based on the above discussion, it will be understood that arbitraryseries and parallel combinations of CAF-n's can be used to generate adesired singleband or multiband frequency response. One example topologyis shown in FIG. 51. In this example, the transfer function of CAF-15101 is:

$\frac{s}{s^{2} + {{.1}s} + 1.4}$

the transfer function of CAF-1 5102 is:

$\frac{s}{s^{2} + {{.1}s} + 1.45}$

the transfer function of CAF-1 5103 is:

$\frac{s}{s^{2} + {{.1}s} + 0.96}$

the transfer function of CAF-1 5104 is:

$\frac{s}{s^{2} + {{.1}s} + 1}$

the transfer function of CAF-1 5105 is:

$\frac{s}{s^{2} + {{.1}s} + 1.02}$

and the transfer function of CAF-1 5106 is:

$\frac{s}{s^{2} + {{.1}s} + 2}$

As will be understood, the frequency response may also be changed byadding level two feedback paths to form signal loops. By applying theprinciples described herein, circuits may be designed that allow for adesired frequency response, and that may be controlled as desired.

Generalized Multi-Pole Filters from CAF SOS Segments

In this section, the application of multiple CAF-n's making up a higherorder bandpass filter will be described. These are intended to beexamples of practical application of the CAF-n's to demonstrate theflexibility of these more atomic components, with the understanding thatnot all possible uses or combinations can be described. For example, oneembodiment may be a larger programmable chip in which a number ofgeneric CAF-n's are built with a switch matrix analogous to ALU's in anFPGA device. This is illustrated in FIG. 52. Here, a plurality of inputand output buffers 102 are attached to a switch matrix 5202, which alsocan connect the input and output of CAF-n 5204 components. Using thisdesign, an arbitrary order bandpass filter can be created by cascadingsuch components as shown in FIG. 53, which is an example of a thirdorder band pass filter, with the various components connected to providea desired signal path 5302.

It will be understood that the examples shown in FIG. 52 and FIG. 53 areexamples of what may be possible using a switch matrix 5202, althoughother configurations are possible. For example, the switch matrix mayhave additional CAF-n elements or other circuit elements that can beconnected in parallel, in series, or in combinations thereof. In thisway, a fabric of CAF-n filters may be customized for specificapplications.

Performance of Generalized Multi-Pole Filters from the CAF SOS Segments

The overall filter transfer function in terms of a rational polynomialin s is now decomposed into SOS's. To avoid the confusion with the SOSinternal to the CAF-n, the consideration here is the overall filterdecomposition into SOS's where the poles of each SOS correspond to adominant pole of the CAF-n. The general higher order filter to beimplemented is given as a rational polynomial in s as:

${H_{bpf}(s)} = \frac{b_{0} + {b_{1}s} + {\ldots\mspace{14mu} b_{M}s^{M}}}{1 + {a_{1}s} + {\ldots\mspace{14mu} a_{M}s^{N}}}$

where M≥N. Here all of the coefficients are real valued. This can beexpressed as a product of L=N/2 SOS sections as follows:

${H_{bpf}(s)} = {\prod\limits_{k = 1}^{L}\;\frac{b_{0,k} + {b_{1,k}s} + {b_{2,k}s^{2}}}{1 + {a_{1,k}s} + {a_{2,k}s^{2}}}}$The poles and zeros are first grouped into complex conjugate pairs.While the combining or grouping of a pair of conjugate poles and pair ofconjugate zeros is somewhat arbitrary, there are generallyimplementation issues that favor some groupings over others. Generallypoles and zeros are grouped that are closer together in a Euclideansense in the S-plane.Example of a CAF-11 Configured as a Butterworth Type Bandpass Filter

A Butterworth bandpass filter will be used as an example which has theform of

${H_{bpf}(s)} = {\frac{{gs}^{N/2}}{1 + {a_{1}s} + {\ldots\mspace{14mu} a_{M}s^{N}}} = {g_{o}{\prod\limits_{k = 1}^{N/2}\;\frac{s}{1 + {a_{1,k}s} + {a_{2,k}s^{2}}}}}}$where g_(o) is a real gain coefficient to be determined as part of theSOS partitioning.

An example of a 4^(th) order Butterworth bandpass filter with a passbandbetween the normalized frequencies of 1 and 1.5 is considered,decomposed into four SOS sections. A desirable feature of theButterworth filter is that it has an optimally flat passband. FIG. 54shows the frequency response of the Butterworth emphasizing thispassband flatness. The left side response curve FIG. 54a is magnified onthe right side FIG. 54b to provide detail in the passband down to −3 dB.

To achieve this response, four SOS sections 5502 a/b/c/d are placed inseries as shown in FIG. 55 and controlled by a controller 5504. Notethat the series connection of the SOS CAF-1 elements 5502 a/b/c/d ispreferred as this avoids having to use parallel connections that have tobe scaled and phased to a precise value. For this example, the CAF-4configuration of FIG. 55 will be used.

The response curve of the normalized gain of the individual CAP-1's,represented by lines 5602 a/b/c/d, is shown in FIG. 56. Note that eachof the four CAF-1 elements are of modest Q, notwithstanding the gainflatness and the steep transition of the fourth order Butterworth. Nextthe frequency response of the cascaded set of four CAF-1's, representedby line 5604, is also shown in FIG. 56, presenting a reasonable match tothe desired Butterworth response, represented by line 5606, with whichwe started. Note that the out of band transition of the four CAF-1'simplementation of this example is actually steeper than that of theoriginal Butterworth filter. This is because of the extra poles in theCAF-1's. However, the shoulders are softer than desired. This may becorrected with a better optimization.

Based on the discussion above and the examples provided, those skilledin the art will recognize that:

1. A wide variety of higher order filters can be realized.

2. Generic filter optimization methods can be used that will configurethe CAF-n's for optimum system level performance. This could be basedon, for example, an eye diagram of an adaptive filter used in acommunication receiver.

3. The Q-spoiler mode may be used in the individual resonators withinthe CAF-n for synthesizing lower Q poles.

Simplified Realizations of SOS Resonator Sections

By way of example, some embodiments of the SOS resonators sections usedin the CAF-n will now be given. These are intended as simplified circuitimplementations of candidate embodiments and not as detailed circuits.

A parallel type implementation of a second order SOS filter element 5702is shown in FIG. 57. There are two control inputs 5704 and 5706, wherecontrol input 5704 is for a variable resistor 5708 which is a Q-spoiler,and control input 5706 is for the variable capacitor 5710, which willcontrol the resonance frequency. The Q-spoiler 5708 moves the resonantpole horizontally along the real axis and the variable capacitor control5710 moves the pole vertically along the imaginary axis.

The Q-spoiler 5708, which moves the pole further into the LHP, isconvenient if more control is desired by the closed loop gain G. FIG. 58shows a parallel resonance circuit 5802 with a variable resistor 5808controlled by control input 5804, and a variable capacitor 5810controlled control input 5806 by the Q-spoiler.

Those skilled in the art will realize that there are a variety of waysthe SOS elements can be implemented and combined. In another embodimentsuitable for the CAF-3, referring to FIG. 59, the three SOS CAF-1resonators may be combined into a single 3^(rd) order bandpass filtertopology consisting of two series resonators 5902 and one parallelresonator 5904. The depiction of resonators 5902 and 5904 has beensimplified and does not include the various control elements, etc.,discussed above. This circuit offers a smaller implementation footprintas the inter-SOS buffers are not required. However, a disadvantage isthat it is harder to tune.

General CAF-n Concept

The first level of generalization is that a number of R resonators 1301can be cascaded as shown in FIG. 60. Here, N such resonators 1301 arecascaded with controls of {f1,G1} to {fN,QN}. Level two feedback may beimplemented across selected adjacent individual resonator elements forimproved performance as described herein.

A feedback path 110 with variable gain G is then around the N cascadedresonators 1301 to form a signal loop as shown in FIG. 61. Severalexamples will be given of useful behaviour of practical significanceprovided by this circuit. The feedback gain is in general complex andreferred to as vector modulation. This implies that the magnitude of Gcan be varied from zero to some maximum value. In addition, the phasecan be varied over 2π radians.

FIG. 62 shows the symbol used to represent a generalized CAF-n 6202 withthe control inputs as {G1, f1, Q1, . . . , GN, fN, QN}.

The next layer of generalization is that several CAF-n's 6202 can becascaded with a feedback loop 110 a with gain as shown in FIG. 63.

As can be seen, there is a general nesting of cascaded Rs 1401 andCAF-n's 6202 with feedback and variable gain. The purpose is to realizearbitrary, multi-pole bandpass filters. These filters are such that theposition of the poles comprising the filter can be moved independentlyand arbitrarily. This is enabled by the variable Q control of each Rblock and the feedback gains denoted by G.

As there is redundancy in the control of the position of the poles, notall the controls need to be independent. Also for practicalimplementations the controls may be coupled together. The selection ofcontrols used and groupings depends on the desired application and alarge variety of such is envisioned. In the following sections a set ofexamples will be given emphasizing a subset of these controls.

Application of the CAF-3 Filter in a Sensor Communications WirelessTransceiver (SWT)

As an illustration of a practical application of the CAF-n filter, ageneric sensor is considered. For this application it is assumed thatthe sensor telemetry transceiver initially has to determine anunoccupied portion of the spectrum to operate within and then contendwith interference. A block diagram of the transceiver is as shown inFIG. 64, A T/R switch block 6402 connects the receiver 6404 and thetransmitter 6404 to the antenna 6408. As the sensor telemetry is lowpower transmission, the CAF-3 6410 can provide the spectral shaping ofthe transmitted signal generated by the transmitter processing block6406. On the receiver side, the CAF-3 6412 filters out interferingsignals and centers a passband on the desired receive signal for thereceiver processing block 6404.

Referring to FIG. 65, as the CAF-3 is frequency agile (that is, thefrequency can quickly be tuned from one passband to another) there is anoption of having only a single CAF-3 6502 that is shared by thetransmitter and receiver functions 6406 and 6404. In the example shown,this is achieved by a circuit that uses a T/R switch 6504 with two portsand a power detector 6506.

CAF-n Based SWT as a Cognitive Radio

An enhanced implementation of the telemetric transceiver for sensorapplications is the cognitive radio aspect of the CAF-3 in which a broadsegment of the spectrum is continuously scanned to monitor activity. Theidea is to find a spectral region in which there is little activity.Finding such a spectral hole, the transceiver can begin datacommunications operations using minimal transmit power and maximalSignal to Noise ratio (SNR) consistent with the environment.

Finding such a spectral hole is achieved by setting up the CAF-3 for aQ-enhanced narrow band filter response that is swept over the potentialfrequency band by varying the f controls of the three Rs in the CAF-3. Asimple power detector at the output of the CAF-3 can be used to estimatethe spectral power at the frequency of the CAF-3. The detector output isthen a function of the applied frequency control for the R's. Aftercompleting a sweep the desirable spectral region(s) can be determined.

Receiver Processing and CAF-3 Control

To integrate the CAF-3 into the transceiver we have to have a means ofproviding a control feedback for stabilizing the approximate position ofthe closed loop poles. This can be done within the forward open loopcontrol by setting the control voltages based on the desired spectrallocation using a reference CAF cell on the same die. The impulseresponse of this reference die would be measured based on a quartzcrystal based time reference that then accounts for variables such astemperature, aging, chip supply voltage, and so forth. An adaptivefeedback is used to precisely tune the CAF-3. This adaptive control canbe implemented without additional circuitry based on using the measuredoutputs of the innate baseband processing already part of the datacommunications receiver.

Start with the standard model of a wireless communications link as shownin FIG. 66. An input data source 6602 is baseband modulated with asymbol or pulse shaping function 6604. This is then up-converted to RFby block 6606 and transmitted over the wireless channel 6608. Thewireless channel 6608 will include standard thermal noise as well asinterference from wireless sources in adjacent channels. The signal isintercepted by the receiver antenna (not shown) then filtered in a CAF-36610, down converted in block 6612 and then a baseband filter 6614 isapplied that matches the pulse shaping filter of the transmitter. Thenthe data is extracted as block 6616. In this data extraction process theSNR of the demodulation process is available which is used in the Hock‘adaptive optimization of CAW-3’ 6618, which has an algorithm 6620 foroptimizing the control settings of a CAF-3 6610. The output of thisblock is converted to voltage control signals in block 6622 for theactual CAF-3 as shown. The objective of the optimizer is to maximize theSNR of the data demodulation process. As the SNR is sensitive to thepole placement of the CAF-3, it gives us a means of continuouslyadapting the control of the CAF 3.

The analysis will be described in terms of equivalent envelopeprocessing. Hence even though the CAF-3 operates at RF we will omit thisby assuming an equivalent baseband model of the CAF-3. Start with thestandard BPSK (binary phase shift keying) modulation format based on aroot raised cosine pulse shaping filter. We could use any arbitrary datacommunication format and modulation as the operation of the CAF-3 is thesame. However, the BPSK avoids unnecessary detail in our presentexplanation.

While the receiver will not have a full eye diagram scope implementationit is useful for illustration here. The processing of the eye diagramscope is done to extract the SNR. We assume that symbol level trackingis done by the receiver to lock the symbol sampling to the point in theeye where it is maximally open.

FIG. 67 depicts an ideal AWGN (added white Gaussian noise) channel 6702,which is connected to a Bernoulli binary generator block 6704, a raisedcosine transmit filter block 6706, a raised cosine receive filter block6708, and a discrete-time eye diagram scope 6710. In this channel, thematched filter is optimum in terms of achieving the maximum eye SNR. Aroot raised cosine (RRC) pulse shaping is used for the transmittedsignal in block 6706 and also for the received signal in block 6708. Inthe case of the ideal AWGN channel, the CAF-3 merely deteriorates theeye unless it is of bandwidth wider than the signal. However, thechannel also has noise from interference from adjacent channels. This isillustrated in FIG. 68, where the AWGN is indicated by 6802, and thepower spectral density (PSD) of the interference signal is indicated by6804. The desired communication channel is indicated by 6806, and theoptimum response of the CAF-3 filter is indicated by 6808.

We now consider the RRC filter which is shown in FIGS. 69a and 69b . Itis an FIR representation of the RRC filter that has 8 samples per symbolepoch and is 10 symbols long. The frequency response of the filter isgiven on the right with respect to half the sampling rate (fundamentalsampling rate is 8 times per symbol).

Next we will assume that there are two sources of noise, the AWGN andthe adjacent channel interference. The power spectral density (PSD) ofthe total noise is given in FIG. 70. Note that the dB scale is relative.

Next we consider the model of the CAF-3. For this discussion werepresent it as an equivalent 3^(rd) order Butterworth low pass filter.Butterworth is selected as it has a very similar pole pattern asproduced by the band-pass CAF-3. The Z transform poles and zeros areshow as the modeling is in discrete time with a sample rate of 8 samplesper symbol epoch. The equivalence is reasonably accurate given therelatively high oversampling rate.

To model the change in the bandwidth of the CAF-3 we will change thebandwidth of the Butterworth LPF by changing the angle of the flankingpoles slightly. For example, the filter above is modeled by setting thebandwidth at 0.44 Hz relative to an 8 Hz normalized sampling rate. InFIG. 71 we have the pole/zero pattern for a CAF-3 filter with anormalized bandwidth of 0.08 Hz with 8 Hz sampling. The center frequencyoffset of the CAF-3 can be represented by a complex discrete time filterby simply rotating the pole zero pattern in FIG. 71 by an amountcommensurate with the frequency offset.

Finally we form the eye diagram by reorganizing the receiver filteredsignal vector. The result is shown in FIG. 72 with the top plot beingthe eye diagram and the bottom plot being the signal to noise ratio(SNR) as a function of sample offset. The SNR is determined by theinverse variance about the two clusters of samples corresponding to thebinary bit demodulation of −1 or 1. Finally one achieves the desiredresult where the symbol clock recovery samples at the point of maximumSNR.

The CAF-3 feedback for the optimization merely uses the maximum SNR asdetermined to adjust the filter controls. It is necessary to use adithering control to do this that varies the three CAF-3 controls:

-   -   1. Q-enhancement/spoiling of the three resonators    -   2. Center frequency of the triplet of resonators    -   3. CAF-3 overall loop level two feedback gain which either        increases Q of center pole and decreases Q of flanking poles        (positive feedback) or decreases Q of center pole and increases        Q of flanking poles (negative feedback).        In this simplified demonstration using a Butterworth filter we        only have two controls:    -   1. Change bandwidth by increasing offset of flanking poles    -   2. Rotate pole zero pattern

FIG. 73 shows the plot of the SNR which has a clear maximum inferring toa normalized bandwidth of 0.1 relative to the symbol rate is optimum.Note that the distortion caused by the CAF-3 which leads to eye closureis offset by the reduction of noise resulting from the largeinterference on either side of the signal passband.

Finally we consider the relative frequency offset of the CAF-3 filter asdescribed earlier. The plot in FIG. 74 shows the SNR of the eye diagramas a function of this offset. Note again that there is a well-definedmaximum in terms of the SNR.

In summary, the examples above demonstrate the ability of using thefeedback from the communications receiver demodulator output into theoptimizer that adjusts the CAF-3 feedback as well as the three CAF-1s.There are of course many different modulation and demodulation schemesas there are many ways of gathering metrics for the optimization of theCAF control. The SNR of the sampling, as shown in this document, issimple and available without further hardware required to the receiver.

General CAF-n Tracking Algorithm Based on Dithering

The general CAF-n will have multiple controls. These can be set in a‘feed forward’ manner as will be described in the next section. However,this assumes that the optimum operating point of the CAF-n is known anda LUT (look up table) maps the desired operating point to physicalvoltages of {f,Q,G}. In the sensor communications tracking problemexample given in the previous section the optimum operating point is notknown and varies depending on the current interference conditions. Thegoal in this example is to optimize the SNR of the receiver output. FromFIGS. 73 and 74, there is clearly at least a local optimum that dependson the two parameters of G and f. In this example the CAF-3 was assumedwith the R controls of f=f1=f2=f3 and Q=Q1=Q2=Q3. Q was set first forthe Rs and then the tracking loop would set G and f. Hence let the SNRof the communications receiver output for a specific instance ofinterference in the power spectral domain be denoted as g(G,f). It isunderstood that g(G,f) will change with time as the spectralinterference is unknown and uncontrollable. The objective is to optimizeg(G,f) which implies finding a stationary point denoted as {f=f_(o),G=G_(o)} where

$\left. {\frac{\partial}{\partial G}{g\left( {G,f} \right)}} \right|_{{f = f_{o}},{G = G_{o}}} = 0$$\left. {\frac{\partial}{\partial f}{g\left( {G,f} \right)}} \right|_{{f = f_{o}},{G = G_{o}}} = 0$If the gradient of g(G,f) was known then the commonly used steepestascent optimizer could be used such that the iteration of G and f wouldfollow

$\begin{bmatrix}f \\G\end{bmatrix}_{new} = {\begin{bmatrix}f \\G\end{bmatrix}_{present} + {\alpha{\nabla{g\left( \begin{bmatrix}f \\G\end{bmatrix}_{present} \right)}}}}$

where α is some small positive real parameter selected based on howquickly the spectral interference changes. It is generally determinedexperimentally. However,

$g\left( \begin{bmatrix}f \\G\end{bmatrix}_{present} \right)$is not generally known to a sufficient degree of accuracy, and therefore

$\nabla{g\left( \begin{bmatrix}f \\G\end{bmatrix}_{present} \right)}$is determined numerically by determining the following three SNR valuesof

$\left\{ {{g_{0} = {g\left( \begin{bmatrix}f \\G\end{bmatrix}_{present} \right)}},{g_{1} = {g\left( {\begin{bmatrix}f \\g\end{bmatrix}_{present} + \begin{bmatrix}{\Delta\; f} \\0\end{bmatrix}} \right)}},{g_{2} = {g\left( {\begin{bmatrix}f \\G\end{bmatrix}_{present} + \begin{bmatrix}0 \\{\Delta\; G}\end{bmatrix}} \right)}}} \right\}$The optimizer then follows the simple algorithm ofif g1>g0 then increase f by Δf otherwise decrease f by Δfif g2>g0 then increase G by ΔG otherwise decrease G by ΔG

In this way the CAF-3 control is always hunting for the optimum SNR. Theincrements of Δf and ΔG depend on the application. This is a form ofdithering control which can be applied to an optimizer when theobjective function is essentially unknown. If the interference changesslowly then Δf and ΔG can be small, however, this requires a longerdwell time to measure the commensurate small change in SNR. If theinterference changes more rapidly then it is necessary to use a largerdithering step size of the parameters to be optimized. Generallydithering schemes are robust and simple to implement hut seldom providethe best performance as it does not take advantage of knowncharacteristics of the objective function.

An Alternate CAF Stability Circuit

In the previous example, the communications receiver output SNR wasoptimized via a dithering algorithm to set the CAF-3 filter controls andto track changes resulting from device drift and un-modeled parametersaffecting the CAF-3 response. Another form of a stability circuit ispresented in this section. FIG. 75 is based on a circuit that hasprocessing built in for the purpose of calibrating and stabilizing theresponse of a CAF-1 block 7502.

The microprocessor 7504, through a DAC implemented as a PWM (pulse widthmodulation) circuit 7506 and based on readings from a temperature sensor7508, adjusts the control for the frequency and Q of the CAF-1 7502. Itdrives this such that the CAF-1 7502 just starts to self-oscillate. Thefrequency of this self-oscillation is down converted in block. 7510 by afrequency synthesizer signal generated by a crystal reference 7505 and afrequency synthesizer 7507 that is set also by the microprocessor 7504.A frequency counter 7509 or other measurement means determines thefrequency of the down-converted signal. In this way the resonantfrequency of the CAF-1 can be determined. Also there is a power detector7512 and ADC block 7514 that can estimate the rate of increase of theself-oscillation signal at the output of the CAF-1 7502. Themicroprocessor 7504 estimates this exponential rise of power and fromthis determines where the closed loop pole of the CAF-1 7502 is.Presently it will be just to the right of the jω axis. If theQ-enhancement is decreased slightly then the self-oscillation willcontinue at the same frequency to a high accuracy but will begin todecay exponentially. No the pole is on the left hand side of the jωaxis. Again based on the power detector 7512, this exponential decay canbe measured and the operating point measured. By repeatedly bringing theCAF-1 7502 into self-oscillation and then reducing the Q-enhancement bya controlled amount the mapping of the CAF-1 7502 to the f and Q controlsignals can be completed. This calibration can be done based oncircuitry on chip that requires no additional off chip components exceptfor the crystal reference source. During operation calibration breakscan be made such that the LUT 7516 is continuously updated. In the caseof the wireless sensor, the transmitter and receiver functions areseparated by epochs of inactivity in which the calibration processingcan be done.

FIG. 76 shows an example of the Q control of the CAF-1 alternatedbetween higher and lower levels that alternately places the closed looppole of the CAF-1 in the right hand and left hand planes. The resultingexponential rise and decay is easily measured by the power detector withnumerical analysis done on the microprocessor. Hence the applied Qcontrol voltage can be mapped to a specific pole position. This isstored in the LUT such that when a pole position is required for thefiltering operations of the CAF-1 then the LUT can be interpolated andthe {f,Q} control voltages set.

The real part of the closed pole value is easily measured. Suppose thatthe pole is in the right hand plane such that it has the form of a exp(bt) where a and b are constants that are unknown. Then if the envelopevoltage is measured at two different times of t₁ and t₂ resulting in x₁and x₂ respectively then the desired parameter b can be estimated from

x₁ = a  exp (bt₁) x₂ = a  exp (bt₂)$b = \frac{\ln\left( \frac{x_{2}}{x_{1}} \right)}{t_{2} - t_{1}}$

Either the times of t₁ and t₂ can be set and the voltages x₁ and x₂measured or else fixed thresholds can be set at x₁ and x₂ and the timedifference of t₂−t₁ measured. Either approach is straight forward.

Other Examples

It will be understood from the forgoing that the CAF-n can be designedto be relatively easy to broaden the bandwidth. As was shown above, asmall amount of feedback gain G around the 3 Rs of a CAF-3 is simple androbust way to change the passband from something representing a singlepole passband response to a broader response similar to a second orderChebyshev.

An application of this is the wireless sensor where the CAF-3 can beconfigured on the fly to be a sensor for power spectral density for acognitive radio, then provide a transmitter fitter for this passband anda receiver filter using T/R switches to reorient the filter for transmitand receive functions. In the receive mode we can tie the CAF-3 into anadaptive loop that uses the SNR of the digital sampled output tooptimize filter parameters. As the {f,Q} control is orthogonal this SNRoptimization is robustly achieved with a simple dithering algorithm.Finally in setting up the CAF-3 it is necessary to know where the polesare located as a function of the controls. This can be achieved by usinga CAF-1 on chip for calibration purposes. By alternating the poleposition in the right and left hand planes (unstable and stablerespectively) the envelope of the self oscillation of the CAF-1 can beused as a probe signal to estimate the real part of the pole location.The imaginary component is determined by the frequency ofself-oscillation. The self-oscillation can be measured based oncomparison with a crystal locked synthesizer frequency. Instead of theCAF-1 the Rs of the CAF-3 can also be used directly. However,implementing the CAF-1 allows for calibration to be continuous and inparallel with the operation of the CAF-3 which is then dedicated for theactual signal processing. The measurements of the CAF-1 pole location asa function of the control voltages and perhaps chip temperature isstored in a LUT. The values of the LUT are interpolated when the CAF-3is to be configured.

Other use examples of the CAF may include, but are not limited to:Equalizer for an antenna to provide a flat antenna response (enhancedfiltenna); Optimized front end module (FEM); Frequency synthesizer;Integrated sensor/cognitive radio system for remote patient monitoring;Low cost sensor transceiver network for Smart roads, Smart signs, andSmart/Driverless cars; Enhanced WiFi systems operating at both 2.4 GHzand 5.0 GHz; Enhanced Bluetooth systems; General high performance analogtunable RF filtering; Spectrum surveillance systems for electronicwarfare applications; Automobile proximity radar modules operating atvery high frequency (above 20 GHz) for collision avoidance; Variable andtunable delay-line modules; Phase control modules; Enhanced sensitivitydetectors for remote sensing applications; Drone surveillance andcontrol;

Phase Shifter

As is known in the art, the phase of a signal may be affected by manydifferent factors as it passes through a circuit, some of which includestray component capacitances and inductances that may be referred to as“parasitics”. As a result, it may be necessary to incorporate a phaseshifter to correct the phase of a signal passing through the circuit.The phase shifter used will depend on the actual implementation of thecircuit. Various types of phase shifters are known in the art, and aperson of ordinary skill may incorporate a suitable phase shifter into aCAF-n as needed. There will now be described some alternative designwhen implementing a phase shifter to the CAF-n circuit, with theunderstanding that it is not possible to described all possibleimplementations.

In general, for a CAF-n circuit that is implemented on a chip, whereparasitics are generally minimal, well modeled and understood, and wherethe CAF-n circuit is intended to be used over a modest frequency range,a fixed phase shifter may be adequate. In other circumstances, it may benecessary to incorporate a variable phase shifter.

It has been found that the closed loop passband of the CAF-1 formsaround the range of frequency where the open loop phase shift is amultiple of 360 degrees. As it is desired to have only a singlepassband, the passband of the resonator may be arranged to coincide withthe frequency of a multiple of 360 degrees phase shift. If the resonatorpeak frequency is misaligned, then the closed loop response peak willstill coincide with the frequency at which a multiple of 360 isachieved, although the passband may be distorted. In addition, it hasbeen found that a detuned resonator adds a phase shift and can be usedto add a controllable modest phase shift. Furthermore, it should berecognized that there is no such thing as a true phase shifter. Anyphase shifter is really an implementation of a variable delay with someassociated magnitude response that is frequency dependent.

Accordingly, a variable phase shift may be introduced by starting with avariable delay line that is made up of a uniform sequence of varactordiodes along a transmission line. By varying the varactor voltage, thegroup delay can be varied, and by changing the group delay, the phasecan be shifted.

As the array of varactors is finite in length and spacing, it will havea non-uniform amplitude response in terms of frequency. In this context,a single varactor will act similar to a low Q resonator with a variablecapacitance, and three varactors will act similar to three coupledresonators as in the CAF-3, Thus, a small number of varactor diodes maybe used to approximate a variable delay line by creating a structurethat has variable group delay with a reasonably uniform magnituderesponse over a desired frequency range. Similarly, 90 degree hybridcouplers used in a quadrature modulator chip are essentially Hilberttransformers that work over a modest frequency range. In other words,variable phase shifters, resonators, delay lines and quadraturemodulators may be considered as circuits arranged and optimized toprovide a variable delay over a range of frequencies.

By generalizing the variable resonator and variable phase shifter andrecognizing that they are functionally similar in the context ofapplication to the CAF-n, it is possible to use a plurality ofsub-circuits in the loop, where each sub-circuit can be controlled togive a desired delay and amplitude response that can be controlled by aplurality of control voltages.

Some specific examples of such sub-circuits are shown in FIG. 77 through79. FIG. 77 shows a variable delay line with a ladder network ofinductors 7702 and capacitors 7704, where the capacitors 7704 arevariable, and may be varactor diodes. Port matching of this sub-circuitgives rise to a magnitude response which may be optimized for a desiredfrequency range. Referring to FIGS. 78 and 79, a series or parallelresonator may be used, which include an inductor 7702, a variablecapacitor 7704, and resistors 7706 as required. FIG. 78 shows a possibletunable delay line having inductors 7702 and capacitors 7704. Thecapacitors 7704 are controllable with a tuning voltage, as representedby the arrows. The more LC sections used, the better the approximationof a variable delay function. FIG. 79 shows a possible phase shifterhaving a parallel resonance circuit with resistors 7706, a variablecapacitor 7704, and an inductor 7702, where the capacitor 7704 isvariable by an applied tuning voltage as represented by the arrow.

FIG. 80 shows a possible implementation of a phase shifter implementedusing a hybrid coupler 8002, which may be described as a distributed orlumped element hybrid coupler, or rat race coupler, where variablecapacitors 7704 are attached to the coupled ports. The phase of coupler8002 is controllable by appropriate adjustments of the capacitors 7704.

Now consider the CAF-1 with a single variable resonator sub-circuit.Potentially, with careful design, the phase shift may be a multiple of360 degrees at a desired frequency within the passband of the resonator.Shifting the resonant frequency equivalently shifts the phase. The CAF-1response peak will occur where the loop phase shift is a multiple of 360degrees. The limitation with the CAF-1 with only a variable resonator isthat the phase shift adjustment of the resonator is limited. Hence ifthe loop has a large phase error, then there is not enough range withthe single resonator, requiring a variable and fixed phase shifter to beadded. However, based on the above discussion, this is equivalent tostringing a number of delay controllable sub-circuits in series.

Now consider substituting another resonator for the variable phaseshifter. The phase shifter has a flatter frequency response in terms ofmagnitude and can therefore be used over a larger frequency range, butthis comes at a cost of adding more components, some of which aredifficult to integrate into a chip. If three resonators are added, thisis equivalent to a CAF-3. This is shown in FIG. 81, with three variableresonators 502, which may be CAF-1 elements, a feedback path 110, acoupler 104, and a gain element 112, which may be controllable.

It will be recognized that there may also be more than three resonators.With more controllable resonators present, more delay adjustment ispossible and this implies a broader frequency range of tuning withouthaving to add fixed phase shifters.

In this patent document, the word “comprising” is used in itsnon-limiting sense to mean that items following the word are included,but items not specifically mentioned are not excluded.

A reference to an element by the indefinite article “a” does not excludethe possibility that more than one of the elements is present, unlessthe context clearly requires that there be one and only one of theelements.

The scope of the following claims should not be limited by the preferredembodiments set forth in the examples above and in the drawings, butshould be given the broadest interpretation consistent with thedescription as a whole.

The invention claimed is:
 1. A variable analog filter, comprising: asignal loop comprising a signal path and a feedback path connectedbetween a signal input and a signal output; and a plurality of circuitelements connected in the signal loop, the plurality of circuit elementscomprising: a plurality of frequency tunable resonators connected inseries; and an adjustable scaling block that applies a gain factor thatis adjustable in a range that comprises a positive gain and a negativegain; and a controller connected to tune the frequency of each frequencytunable resonator and to adjust the gain factor of the adjustablescaling block toward a desired frequency response.
 2. The variableanalog filter of claim 1, wherein the frequency tunable resonators andthe adjustable scaling block are connected in series in the signal pathof the signal loop.
 3. The variable analog filter of claim 1, whereinthe controller adjusts the frequency tunable resonators to achieve adesired phase shift in the signal loop.
 4. The variable analog filter ofclaim 1, wherein two or more frequency tunable resonators are connectedin primary signal loops, the primary signal loops being connected inseries within the signal loop, each primary signal loop comprising aprimary adjustable scaling block that is controlled by the controllerbetween a negative gain factor and a positive gain factor.
 5. Thevariable analog filter of claim 1, wherein the controller is programmedto selectively Q-spoil or Q-enhance one or more frequency tunableresonators toward the desired frequency response.
 6. The variable analogfilter of claim 5, wherein the controller is programmed to Q-spoil theone or more frequency tunable resonators by applying a negative gainfactor, and Q-enhance the one or more frequency tunable resonators byapplying a non-negative gain factor.
 7. A multipole filter comprising aplurality of variable analog filters of claim 1 connected in series, inparallel, or combinations thereof.
 8. A method of filtering an analogelectromagnetic signal, the method comprising the steps of: providing afilter comprising: a signal loop comprising a signal path and a feedbackpath connected between a signal input and a signal output; and aplurality of circuit elements connected in the signal loop, theplurality of circuit elements comprising: a plurality of frequencytunable resonators connected in series; and an adjustable scaling blockthat applies a gain factor that is adjustable in a range that comprisesa positive gain and a negative gain; and tuning the frequency of thefrequency tunable resonators and adjusting the gain factor of theadjustable scaling block toward a desired frequency response.
 9. Themethod of claim 8, further comprising the step of controlling a phaseshift within the signal loop by adjusting the frequency tunableresonators.
 10. The method of claim 9, wherein controlling the phaseshift comprises detuning the frequency tunable resonators from a centralfrequency of the filter.
 11. The method of claim 8, wherein thefrequency tunable resonators and the adjustable scaling block areconnected in series in the signal path of the signal loop.
 12. Themethod of claim 8, wherein two or more frequency tunable resonators areconnected in primary signal loops, the primary signal loops beingconnected in series within the signal loop, each primary signal loopcomprising a primary adjustable scaling block that is adjusted between anegative gain factor and a positive gain factor.
 13. The method of claim8, wherein the frequency tunable resonators are selectively Q-spoiled orQ-enhanced toward the desired frequency response.
 14. The method ofclaim 13, wherein the frequency tunable resonators are Q-spoiled byapplying a negative gain factor or Q-enhanced by applying a non-negativegain factor.